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# Walsh function

About: Walsh function is a research topic. Over the lifetime, 1356 publications have been published within this topic receiving 18035 citations.

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560 citations

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TL;DR: It is shown that a Boolean combining function f(x) of n variables is mth-order correlation-immune if and only if its Walsh transform F( omega ) vanishes for all omega with Hamming weight between 1 and m, inclusive.

Abstract: It is shown that a Boolean combining function f(x) of n variables is mth-order correlation-immune if and only if its Walsh transform F( omega ) vanishes for all omega with Hamming weight between 1 and m, inclusive. This result is used to extend slightly Siegenthaler's (IEEE Trans. Comput., vol. C-34, pp. 81-85, Jan. 1985) characterization of the algebraic normal form of correlation-immune combining functions. >

423 citations

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TL;DR: Hadamard matrices have been widely studied in the literature and many of their applications can be found in this paper, e.g., incomplete block designs, Youden designs, orthogonal $F$-square designs, optimal saturated resolution III (SRSIII), optimal weighing designs, maximal sets of pairwise independent random variables with uniform measure, error correcting and detecting codes, Walsh functions, and other mathematical and statistical objects.

Abstract: An $n \times n$ matrix $H$ with all its entries $+1$ and $-1$ is Hadamard if $HH' = nI$. It is well known that $n$ must be 1, 2 or a multiple of 4 for such a matrix to exist, but is not known whether Hadamard matrices exist for every $n$ which is a multiple of 4. The smallest order for which a Hadamard matrix has not been constructed is (as of 1977) 268. Research in the area of Hadamard matrices and their applications has steadily and rapidly grown, especially during the last three decades. These matrices can be transformed to produce incomplete block designs, $t$-designs, Youden designs, orthogonal $F$-square designs, optimal saturated resolution III designs, optimal weighing designs, maximal sets of pairwise independent random variables with uniform measure, error correcting and detecting codes, Walsh functions, and other mathematical and statistical objects. In this paper we survey the existence of Hadamard matrices and many of their applications.

288 citations

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TL;DR: Optimal pulses designed to minimize multiple-access interference in quasi-synchronous systems are obtained for various bandwidths and are shown to provide a large improvement over the raised cosine pulses.

Abstract: Proposes a multicarrier orthogonal CDMA signaling scheme for a multiple-access communication system, such as the reverse channel of a cellular network, as an alternative to the multi-user interference cancellation approach. The average variance of cross-correlations between sequences is used as a measure for sequence design. The authors search for sets of sequences that minimize the probability of symbol detection error, given that there is imperfect synchronization among the signals, that is, the signals are quasi-synchronous. Orthogonal sequences based on the Sylvester-type Hadamard matrices (Walsh functions) are shown to provide a significant improvement over the case where a Hadamard (orthogonal) matrix is chosen at random. Computer searches suggest that this set of codes is optimal with respect to the above measure. The issue of chip pulse shaping is investigated. Optimal pulses designed to minimize multiple-access interference in quasi-synchronous systems are obtained for various bandwidths and are shown to provide a large improvement over the raised cosine pulses. A multicarrier signaling scheme is introduced in order to reduce chip level synchronization offsets between the users. >

285 citations